1. 1 Economics & Evolution Number 2

2. 2 Reading List

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4. 4 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time is spent in each quadrant

5. 5 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time is spent in each quadrant

6. 6 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time spent in the first quadrant analogously:

7. 7 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time spent in the second quadrant And at t 2 :

8. 8 AB A 0, 23, 0 B 2, 11, 3 0 1 1 An Example (B, A ) (B, B ) (A, B ) (A, A ) time spent in the second quadrant (slide 2) (slide 4) Spending the same time in each quadrant

9. 9 Fictitious Play: Failure to Converge An example by L. Shapley LCR T0, 00, 11, 0 M 0, 00, 1 B 1, 00, 0 The only Nash equilibrium is (⅓, ⅓, ⅓)

10. 10 Fictitious Play: Failure to Converge An example by L. Shapley LCR T0, 00, 11, 0 M 0, 00, 1 B 1, 00, 0 T p 1 =1 M p 2 =1 B p 3 =1 L C R L q 1 =1 C q 2 =1 R q 3 =1 T M B History of player 1 (in player’s 2 mind) History of player 2 (in player’s 1 mind) Best Response of 2 Best Response of 1 The process does not converge, (spends longer periods in any part)

11. 11 Fictitious Play: Failure to Converge An example by L. Shapley LCR T0, 00, 11, 0 M 0, 00, 1 B 1, 00, 0 T p 1 =1 M p 2 =1 B p 3 =1 L C R L q 1 =1 C q 2 =1 R q 3 =1 T M B History of player 1 (in player’s 2 mind) History of player 2 (in player’s 1 mind) Best Response of 2 Best Response of 1 They (almost) always play either (0,1) or (1,0)

12. 12 Fictitious Play: Failure to Converge An example by L. Shapley LCR T0, 00, 11, 0 M 0, 00, 1 B 1, 00, 0 T p 1 =1 M p 2 =1 B p 3 =1 L C R L q 1 =1 C q 2 =1 R q 3 =1 T M B History of player 1 (in player’s 2 mind) History of player 2 (in player’s 1 mind) Best Response of 2 Best Response of 1 Exercise: What happens when they start at the following points ???

13. 13 A proof of the non convergence of Fictitious Play in the Shapley game. Monderer, Samet, Sela: ‘Belief Affirming in Learning Processes’ JET, vol 73, April 1997

14. 14 etc. ex anteex post

15. 15 If the process converges it must be that: Impossible !!!! So the process cannot converge.

16. 16 A Social Interpretation to Fictitious play dynamics Two populations of size N, meet at random and play a 2X2 game. At time t, p(t) of the first population play the second strategy [ q(t) for the second population] Players die and are replaced by new ones. The newly born learn to play the best response against the other population at the time of their birth.

17. 17 AB A 0, 23, 0 B 2, 11, 3 The game: (same as in the previous fictitious game example)

18. 18 AB A 0, 23, 0 B 2, 11, 3 q 0 1 (A) (B) (A) p 1 (B) As long as ( p(t), q(t) ) i s in the first quadrant, the best responses are: ( B, A ).

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21. 21 The Replicator Dynamics and Evolutionarily Stable Strategies E.S.S. Replicator: gene, phenotype It replicates, according to how well it did. It determines the behaviour, the strategy. The replicators play, replicate and then die so that the population remains of a fixed size.

22. 22 DH D 1,10, 2 H 2, 0-1, -1 Chicken game, or Dove & Hawk The population plays: 1-p strategy D, and p strategy H. The fitness of a player (the no. of his offsprings): In a short time interval τ : the total number of D,H players after replication: H players:

23. 23 The new proportion of H players:

24. 24 We have not used the particular numbers of the Chicken game. The above equation holds for all 2x2 games, And it is independent of U. The Replicator Dynamics Now apply it to Chicken: DH D 1,10, 2 H 2, 0-1, -1

25. 25 DH D 1,10, 2 H 2, 0-1, -1 t 1/2 D H p < 1/2 p

26. 26 CD C 2,20, 3 D 3, 01, 1 t C D The Prisoners’ Dilemma p

27. 27 LR L 1,10, 0 R 2, 2 t L R p < 1/3 Coordination Game p 1/3 p > 1/3 Spontaneous Order, (no one maximizes)